Rosenbaum Sensitivity Analysis for Fisher's Exact Test

Description

Calculates sensitivity to hidden bias for Fisher's exact test for a two-by-two contingency table, following the method described in Rosenbaum (2002, sec. 4.4).

Usage

FisherSens(totalN, treatedN, totalSuccesses, treatedSuccesses, Gammas)

Arguments

Value

Returns a matrix with three columns and number of rows equal to the length of "Gammas". Each row indicates the upper and lower bounds for the (one-sided) p-value for a given value of Gamma.

Author(s)

Devin Caughey, MIT, caughey@mit.edu

See Also

See also binarysens, hlsens, mcontrol

Examples

## Fisher's Lady Tasting Tea: milk first or tea first?
LadyTastingTea <- matrix(c(4, 0, 0, 4), nrow = 2,
                         dimnames = list(Guess = c("Milk", "Tea"),
                                         Truth = c("Milk", "Tea")))
## Define "Milk" as "treated"/"success"
FisherSens(totalN = sum(LadyTastingTea), 
           treatedN = sum(LadyTastingTea["Milk", ]),
           totalSuccesses = sum(LadyTastingTea[, "Milk"]), 
           treatedSuccesses = sum(LadyTastingTea["Milk", "Milk"]), 
           Gammas = seq(1, 2, .2))
## Interpretation: Rejection of the null hypothesis 
## (that the lady cannot discriminate between milk-first and tea-first) 
## is insensitive to bias as large as Gamma = 2.

Rosenbaum Sensitivity Analysis for Unmatched Groups

Description

Calculates sensitivity to hidden bias for tests based on sum statistics (e.g., Wilcoxon's rank sum test), following the method described by Rosenbaum (2002, sec. 4.6). It is meant for unmatched/unstratified data with ordinal or continuous responses.

Usage

SumTestSens(T, q, n, m, Gamma)

Arguments

Value

This function prints the upper bound of the normal approximation one-sided p-value for the test at the given value of Gamma. It also invisibly returns a list of intermediate statistics.

Warning

Since ‘SumTestSens’ calculates through enumeration the exact expectation and variance of the test under the null, it is very computationally intensive and may be unworkable for even medium-sized datasets.

Author(s)

Devin Caughey, MIT, caughey@mit.edu

References

Paul R. Rosenbaum. Observational Studies. Springer, New York, 2nd edition, 2002, sec. 4.6

See Also

See also binarysens, hlsens, mcontrol

Examples

## Example from Rosenbaum (2002, p.~146)
  mercury <- data.frame(matrix(c(1, 0, 2.7,    5.3,
                                 2, 0, 0.5,   15.0,
                                 3, 0, 0.0,   11.0,
                                 4, 0, 0.0,    5.8,
                                 5, 0, 5.0,   17.0,
                                 6, 0, 0.0,    7.0,
                                 7, 0, 0.0,    8.5,
                                 8, 0, 1.3,    9.4,
                                 9, 0, 0.0,    7.8,
                                10, 0, 1.8,   12.0,
                                11, 0, 0.0,    8.7,
                                12, 0, 0.0,    4.0,
                                13, 0, 1.0,    3.0,
                                14, 0, 1.8,   12.2,
                                15, 0, 0.0,    6.1,
                                16, 0, 3.1,   10.2,
                                17, 1, 0.7,  100.0,
                                18, 1, 4.6,   70.0,
                                19, 1, 0.0,  196.0,
                                20, 1, 1.7,   69.0,
                                21, 1, 5.2,  370.0,
                                22, 1, 0.0,  270.0,
                                23, 1, 5.0,  150.0,
                                24, 1, 9.5,   60.0,
                                25, 1, 2.0,  330.0,
                                26, 1, 3.0, 1100.0,
                                27, 1, 1.0,   40.0,
                                28, 1, 3.5,  100.0,
                                29, 1, 2.0,   70.0,
                                30, 1, 5.0,  150.0,
                                31, 1, 5.5,  200.0,
                                32, 1, 2.0,  304.0,
                                33, 1, 3.0,  236.0,
                                34, 1, 4.0,  178.0,
                                35, 1, 0.0,   41.0,
                                36, 1, 2.0,  120.0,
                                37, 1, 2.2,  330.0,
                                38, 1, 0.0,   62.0,
                                39, 1, 2.0,   12.8),
                               nrow = 39, ncol = 4, byrow = TRUE))
  colnames(mercury) <- c("ID", "Tr", "Pct.cu.cells", "Hg.in.blood")
  
  (T_test <- rank(mercury$Hg.in.blood) %*% mercury$Tr)
  (q_test <- sort(rank(mercury$Hg.in.blood), decreasing = TRUE))
  (n_test <- nrow(mercury))
  (m_test <- sum(mercury$Tr))

  ## Note: since this function uses exact rather than approximate
  ## formulas for the mean and variance of T, the p-values it
  ## calculates do not precisely match those in Rosenbaum (2002).

  #A single Gamma value - example not run
  #testOut2 <- SumTestSens(T = T_test,
  #                         q = q_test,
  #                         n = n_test,
  #                         m = m_test,
  #                         Gamma = 35)
 
  ## Apply to vector of Gamma values
  sapply(c(1, 5, 35), SumTestSens,
         T = T_test, q = q_test, n = n_test, m = m_test) 

Angrist and Lavy (1999) Maimonides' Rule Data Set

Description

Angrist and Lavy (1999) data set used by Rosenbaum (2010) to demonstrate his instrumental variable sensitivity analysis.

Usage

data(angristlavy)

Format

A data.frame with 172 observations on the following variables).

clasz: Size of class or classes for each cohort.

avgmath: Average math test score for each class.

z: A recode of enrollment with 1 indicating a cohort with 41 or more students, i.e. two classes. This serves as the instrument which encourages smaller classes.

pair: matched pair id

Source

Angrist, Joshua and Lavy, Victor (1999). “Using Maimonides' Rule to Estimate the Effect of Class Size on Scholastic Achievement.” Quarterly Journal of Economics 114, 533–575.

Rosenbaum, Paul R. (2010). Design of Observational Studies. Springer-Verlag.

References

Rosenbaum, Paul R. (2010). Design of Observational Studies. Springer-Verlag.

Sensitivity Test for Matched Binary Outcome

Description

Function to calculate Rosenbaum bounds for binary data.

Usage

 
binarysens(x,y, Gamma=6, GammaInc=1)

Arguments

Author(s)

Luke Keele, University of Pennsylvania, luke.keele@gmail.com

References

Rosenbaum, Paul R. (2002) Observational Studies. Springer-Verlag.

See Also

See also psens, hlsens, mcontrol

Examples

 
# Example From Rosenbaum Observational Studies Pg 112
# Success: Died From Lung Cancer
# 110 Discrepant Pairs
# 12 Discrepant Pairs

# Sensitivity Test
binarysens(12,110)

Rosenbaum Bounds for Hodges-Lehmann Point Estimate

Description

Function to calculate Rosenbaum bounds for continuous or ordinal outcomes based on Hodges-Lehmann point estimate.

Usage

# Default Method 
hlsens(x, y, pr = 0.1, Gamma = 6, GammaInc = 1)

Arguments

Details

For large data sets this function can be quite slow if pr is set to low. If the data set is larger, it is best to set pr to .5 before trying values such as .01. Generally, the results from the function are insensitive to the value for pr.

Author(s)

Luke Keele, University of Pennsylvania, luke.keele@gmail.com

References

Rosenbaum, Paul R. (2002) Observational Studies. Springer-Verlag.

See Also

See also binarysens, psens, mcontrol

Examples

# Replication of Rosenbaum Sensitivity Tests From Chapter 4 of
# Observational Studies

# Data: Matched Data of Lead Blood Levels in Children
trt <- c(38, 23, 41, 18, 37, 36, 23, 62, 31, 34, 24, 14, 21, 17, 16, 20,
15, 10, 45, 39, 22, 35, 49, 48, 44, 35, 43, 39, 34, 13, 73, 25, 27)

ctrl <- c(16, 18, 18, 24, 19, 11, 10, 15, 16, 18, 18, 13, 19, 10, 16,
16, 24, 13, 9, 14, 21, 19, 7, 18, 19, 12, 11, 22, 25, 16, 13, 11, 13)

hlsens(trt, ctrl)

Function to calculate Rosenbaum bounds for IV Estimator based on Wilcoxon sign rank test.

Description

iv_sens performs a non-parametric, instrumental variable sensitivity analysis on matched pairs following the logic of the Neyman-Rubin framework for causal inference. The function supports a variable-valued instrument.

Usage

iv_sens(Rt, Rc, Dt, Dc, Gamma = 6, GammaInc = 1)

Arguments

Details

Given matched pairs of observations on an instrument Z, which encourages dose D, this function performs a Rosenbaum's bounds sensitivity analysis. Note that matching is done on levels of the instrument.

Value

Returns an object of class rbounds.

Author(s)

Luke Keele, University of Pennsylvania, luke.keele@gmail.com

References

Angrist, Joshua D., Imbens, Guido W., and Rubin, Donald B. (1996). "Identification of Causal Effects Using Instrumental Variables." Journal of the American Statistical Association 91/434, pp. 444–455.

Rosenbaum, Paul R. (1996). "Comment." Journal of the American Statistical Association 91/434, pp. 465–468.

Rosenbaum, Paul R. (2002). Observational Studies. Springer-Verlag.

Rosenbaum, Paul R. (2010). Design of Observational Studies. Springer-Verlag.

See Also

See also binarysens, hlsens, mcontrol

Examples

data(angristlavy)

# Example from Ch 5 of Design of Observational Studies
	
#Extract Matched Outome Data
Rt <- angristlavy$avgmath[angristlavy$z==1]
Rc <- angristlavy$avgmath[angristlavy$z==0]

#Extract Matched Doses
#Doses Encouraged By Instrument - Here Class Size
Dt <- angristlavy$clasz[angristlavy$z==1]
Dc <- angristlavy$clasz[angristlavy$z==0]


#Run Sensitivity Analsyis - Rank Sum Test
iv_sens(Rc, Rt, Dc, Dt, Gamma=1.5, GammaInc=.01)

Sensitivity Analysis For Multiple Matched Controls

Description

Function to calculate Rosenbaum bounds for continuous or ordinal outcomes based on Wilcoxon sign rank test p-value when there are multiple matched control units.

Usage

# Default Method 
mcontrol(y, grp.id, treat.id, group.size = 3, Gamma = 4, GammaInc = 1)

Arguments

Details

The matched data needs to be in a very particular form for this function to work. The data must be sorted by matched groups with indicators for each matched group and for treated and control units. The simplest way to prepare the data is to use the Match() function and use the data.prep() function to format the data.

Currently this function only takes matched data with 2 or 3 controls units matched to each treated unit.

This function does cannot handle data where the number of control units is not the same for every treated unit.

Author(s)

Luke Keele, University of Pennsylvania, luke.keele@gmail.com

References

Rosenbaum, Paul R. (2002) Observational Studies. Springer-Verlag.

See Also

See also binarysens, psens, hlsens

Examples

 

grp <- c(1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 
         6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10,10,10,11,
         11,11,12,12,12)
trt <- rep(c(1,0,0), 12)
score <- c(-11.39,-8.45, -19.57,-8.33, 3.06, -19.93,-18.73,-11.99,-7.55, 
            11.94, 9.4, -25.16,-0.77, -10.46,-7.27, 24.03, -8.23, 2.67,  
            -4.04, -6.67, -1.12, -14.4, -26.21,5, -1.7, -15.3, -7.73, 
            -0.87, -19.71,-12.69, -3.36, -11.21,-35.83,5.89, -10.79,2)


mcontrol(score, grp, trt, group.size=3)

Summarize output from rbounds object

Description

Summary method for rbounds object.

Usage

## S3 method for class 'rbounds'
print(x, ...)

Arguments

Author(s)

Jason W. Morgan, Ohio State University, morgan.746@osu.edu

See Also

Also see binarysens, psens, and hlsens.

Rosenbaum Bounds for Sign Rank

Description

Function to calculate Rosenbaum bounds for continuous or ordinal outcomes based on Wilcoxon sign rank test.

Usage

# Default Method 
psens(x, y, Gamma = 6, GammaInc = 1)

Arguments

Author(s)

Luke Keele, University of Pennsylvania, luke.keele@gmail.com

References

Rosenbaum, Paul R. (2002) Observational Studies. Springer-Verlag.

See Also

See also binarysens, hlsens, mcontrol

Examples

 
# Replication of Rosenbaum Sensitivity Tests From Chapter 4 of
# Observational Studies

# Data:  Matched Data of Lead Blood Levels in Children
trt <- c(38, 23, 41, 18, 37, 36, 23, 62, 31, 34, 24, 14, 21, 17, 16, 20,
         15, 10, 45, 39, 22, 35, 49, 48, 44, 35, 43, 39, 34, 13, 73, 25,
         27)
ctrl <- c(16, 18, 18, 24, 19, 11, 10, 15, 16, 18, 18, 13, 19, 10, 16,
          16, 24, 13, 9, 14, 21, 19, 7, 18, 19, 12, 11, 22, 25, 16, 13,
          11, 13)

psens(trt, ctrl)